Existence of positive entire radial solutions to a $(k_1,k_2)$-Hessian systems with convection terms

Abstract

In this article, we prove two new results on the existence of positive entire large and bounded radial solutions for nonlinear system with gradient terms $$\displaylines{ S_{k_1}(\lambda (D^{2}u_1) )+b_1(| x| ) | \nabla u_1|^{k_1} =p_1(| x| ) f_1(u_1,u_2) \quad\text{for }x\in \mathbb{R}^{N}, \cr S_{k_2}(\lambda (D^{2}u_2) ) +b_2(| x| ) | \nabla u_2|^{k_2} =p_2(| x| ) f_2(u_1,u_2) \quad\text{for }x\in \mathbb{R}^{N}, }$$ where $S_{k_i}(\lambda (D^{2}u_i) ) $ is the $k_i$-Hessian operator, $b_1,p_1, f_1, b_2, p_2,f_2$ are continuous functions satisfying certain properties. Our results expand those by Zhang and Zhou [23]. The main difficulty in dealing with our system is the presence of the convection term.

Keywords

• Entire solution
• large solution
• elliptic system